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Modeling 9/11 disaster networks using degree sequence models

Miruna Petrescu-Prahova, University of Washington Michael Schweinberger, Pennsylvania State University Duy Quang Vu, Pennsylvania State University June 3, 2011

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

9/11 organizational collaboration network

Response to World Trade Center attacks Data on interorganizational collaboration during the first 12 days after September 11, 2001 was collected from newspapers, field documents, and interviews Organizational attributes: type of organization (e.g., non-profit, governmental, etc.) and scale of the organization (e.g., local, state, etc.) Result: network of collaborations among 717 organizations

226 isolates 10 small components of sizes 2—10

• ne large component of size 463

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

9/11 organizational collaboration network

local state national international collective government non-profit profit

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

9/11 organizational collaboration network: challenges

Degree distribution: small number of high-degree nodes (coordinators) and large number of low-degree nodes ⇒ communication (coordination) depends on small number of nodes.

20 40 60 80 0.0 0.1 0.2 0.3 0.4 0.5 Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Modeling

Fundamental feature of collaboration network: sequence of degrees g1(y), . . . , gn(y), where gi(y) = n

j=i yij.

Maximum entropy principle (Jaynes, 1957a,b) suggests following family

• f exponential-family models:

Pθ(Y = y) = exp n

• i=1

θigi(Y ) − A(θ)

• =

exp  

n

• i<j

(θi + θj)yij − A(θ)   , where A(θ) =

n

• i<j

log [1 + exp (θi + θj)] .

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Modeling and learning

Modeling: Simple starting point. Motivated by data. Dyad-independence exponential-family random graph model. Natural relative of p1 model of Holland and Leinhardt (1981), the first exponential-family random graph model. Can be extended to include covariates. Learning: Bayesian approach with truncated Dirichlet process prior and hyper-prior distribution: Bayesian approach to stochastic block model. Posterior distribution approximated by variational methods as well as MCMC.

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Outline of analyses

Fit ERGMs that include both dyad-independent and dyad-dependent terms. Select number of blocks using variational method Fit stochastic block models that include dyad-independent terms. Compare the goodness-of-fit of the stochastic block models and ERGMs. Analyze composition of blocks

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

ERGMs

Terms: edges, nodemix, 2-star, gwdegree. Covariates: organizational type (4 categories) and organizational scale of operations (4 categories). Goodness-of-fit: compare the properties of the observed network with same properties of simulated networks. M1: edges, nodemix M2: edges, nodemix, 2-star M3: edges, nodemix, gwdegree

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Stochastic block models

Terms: edges and nodemix. Covariates: organizational type (4 categories) and organizational scale of operations (4 categories). First step: determining the best fit in terms of number of blocks using variational EM.

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Selecting the number of blocks

• 2

4 6 8 10 −7200 −7000 −6800 −6600 −6400 #blocks log P(Y = y)

M4: stochastic block model with 4 blocks Primary, secondary and tertiary hubs and near-isolates (Butts et al, 2007).

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

ERGM comparison

M1

#components −200 200 0.00 0.02 0.04

M2

#components −200 200 0.00 0.10 0.20 0.30

M3

#components −400 400 0.000 0.015 0.030

M4

#components −40 40 0.00 0.02 0.04

M1

max(component size) −200 200 0.00 0.02 0.04

M2

max(component size) −200 200 0.00 0.10 0.20 0.30

M3

max(component size) −400 400 0.00 0.04 0.08

M4

max(component size) −40 20 0.00 0.02 0.04

• 1

3 5 7 9 100000 200000

M1

distance

• ●
• ● ● ●
• 1

3 5 7 9 100000 200000

M2

distance

• ●
• ● ● ●
• 1

3 5 7 9 100000 200000

M3

distance

• ●
• ● ● ●
• 1

3 5 7 9 100000 200000

M4

distance

• ●
• ● ● ●

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

ERGM comparison

M1

#2−stars −30000 20000 0e+00 4e−04 8e−04

M2

#2−stars −20000 20000 0e+00 6e−04

M3

#2−stars −30000 30000 0.000 0.015 0.030

M4

#2−stars −5000 5000 0.00000 0.00015

M1

#triangles −1500 1000 0.00 0.03 0.06

M2

#triangles −1500 1000 0.00 0.10

M3

#triangles −1500 1000 5 10 15

M4

#triangles −600 400 0.000 0.002 0.004

• 1

21 44 67 90 50 150 250

M1

degree

• 1

21 44 67 90 50 150 250

M2

degree

• 1

21 44 67 90 50 150 250

M3

degree

• 1

21 44 67 90 50 150 250

M4

degree

• Miruna Petrescu-Prahova, University of Washington

, Michael Schweinberger, Pennsylvania State University Disaster Networks

Block-dependent degree parameters

Table: Posterior median and 95% credibility intervals of hyper-parameters α, µ, and σ−2 and degree parameters θ1, θ2, θ3, θ4

median 95% credibility interval scaling parameter α 1.007 (.298, 2.384) mean parameter µ

• 2.187

(-3.590,-.721) precision parameter σ−2 .537 ( .217, 1.074) degree parameter θ1 .056 (-0.148, .299) degree parameter θ2

• 1.382

(-1.602,-1.114) degree parameter θ3

• 2.739

(-2.937,-2.495) degree parameter θ4

• 4.855

(-5.112,-4.562)

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Primary Hubs

Table: Organizations which are predicted to be primary hubs

Organization Degree Entropy CDC 91 < .001 CMS 79 < .001 DHHS 84 < .001 ESF9 56 < .001 FDNY 33 < .001 FEMA 68 < .001 HCFA 76 < .001 NYC 41 .002 NYCEOC 40 .006 NYCOEM 68 < .001 NYPD 40 .005 PANYNJ 26 .013 VER 33 < .001

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

Discussion

Number of triangles not represented well by the model Prominence of organizations

Formal and emergent hubs Data sources

Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks

References

Holland, P. W., and Leinhardt, S. (1981), “An Exponential Family Of Probability Distributions For Directed Graphs,” Journal of the American Statistical Association, 76, 33–65. Jaynes, E. T. (1957a), “Information Theory and Statistical Mechanics,” Physical Review, 106, 620–630. — (1957b), “Information Theory and Statistical Mechanics II,” Physical Review, 108, 171–190. Miruna Petrescu-Prahova, University of Washington , Michael Schweinberger, Pennsylvania State University Disaster Networks